### Mathematica Slovaca

Goutam Mukherjee; Parameswaran Sankaran

Minimal models of oriented Grassmannians and applications

*Mathematica Slovaca, Vol. 50 (2000), No. 5, 567--579*
Persistent URL:http://dml.cz/dmlcz/136790

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### Slovaca

**©2000 **

**Matli. Slovaca, 50 (2000), No. 5, 567-579 s . ^ l ™ ^ SaSE?.. **

### MINIMAL MODELS OF O R I E N T E D G R A S S M A N N I A N S AND A P P L I C A T I O N S

G O U T A M M U K H E R J E E * — PARAMESWARAN S A N K A R A N * *
*(Communicated by Julius Korbas) *

A B S T R A C T . We construct t h e minimal models for t h e oriented G r a s s m a n n
*manifold G**n k** of all oriented k dimensional vector subspaces of lR*n a n d ver-
ify t h a t they are formal. As an application we obtain a classification of real flag
manifolds according t o nilpotence, which was first established by H. Glover and
W . Homer. We also establish a result of K. Varadarajan t h a t t h e classifying space
*BO(k) is nilpotent if and only if k is odd. *

### 1. Introduction

The purpose of this paper is to give an explicit description of minimal models
of oriented Grassmann manifolds. The construction of minimal models of com-
*pact simply connected homogeneous manifolds is well understood from the work *
of S u l l i v a n [15] and others. (Cf. [4], [7].) However, we have not been able
to find explicit reference for the description, depending only on the parame-
*ters n and k, 1 < k < n, of minimal model of an oriented Grassmann man-*
*ifold G**n k** of oriented k-vector subspaces of R*n. It is our hope that such a
description will be useful in answering many questions about Grassmannians
(oriented as well as unoriented). Using our description of the minimal model,
*we prove that G**n k** is formal. Of course, this is a well-known result since *
the oriented Grassmann manifolds are Riemannian symmetric spaces (see [15;

*p. 326], [8; p. 158] and [9]). (Cf. Remark 2 below.) We apply our results to *
*show that the action of the fundamental group of G**n k*, the Grassmann mani-
*fold of k planes in R**n**, on n**k**(G**nk**) is not nilpotent in case k is even. We *
deduce a result of H. G l o v e r and W. H o m e r that the real flag mani-
*fold G(n**l**,... ,n**s**) = 0(^/(0^) x ••• x 0(n**s**)), n = J2 n**{* is not nilpo-

*l<i<s *

*tent when one of the n**i* is even. We also obtain a new proof of a result of

2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : P r i m a r y 55P62, 55P99.

*K e y w o r d s : G r a s s m a n n manifold, flag manifold, rational homotopy t h e o r y minimal model, *
*formality, nilpotence. *

*K. V a r a d a r a j a n that the classifying space BO(k) is nilpotent if and only *
*if k is odd. *

*We hope that our method of explicit construction and proof, more than the *
*results themselves, will be of some interest. Our approach to nilpotence via the *
*theory of minimal models is probably new. *

*We now state the main result of this paper. Write k = 2s or 25 + 1 , n — k = 2t *
**or 2t+ 1, 1 < s,t e Z , n > 2k. **

*Let P denote the polynomial algebra R[p**l**,... ,p**s**], where each p• is homo-*
*geneous of degree |p 1=4,7. Define homogeneous elements /i £ P of degree 4j *
*by the equation (1 +p**x** + • • • +p**s**)(l + h**Y** + • • • + hj + •••) = 1 . Thus hj , j > 1, *
*is a certain polynomial in p**l**,..., p**s**. *

*Introduce elements o**k**, r**n**_**k** of degree k and n — k such that o\ = p**s* if
*k is even, o**k** = 0 if k is odd; T**n**_**k** = 0 if n — k is odd, otherwise it is an *
*indeterminate. Let A be the algebra got by adjoining o**k**, T**n**_**k** to P. Note that *
*A is a polynomial algebra in even degree generators. *

*Let M := M**nk* denote the commutative differential graded algebra over the
*reals defined as follows. Recall that commutativity is in the graded sense: for *
*homogeneous elements u and v, uv = ( — l ^ l ^ W . *

*Case 1: *

**Let n = 2m, k = 2s, n — k = 2t, s > t. Let M = A[u****0****,v****0****,... ,v****s****_****l****], \vj\ = ***4(£ + j) — 1, 0 < j < s, \u**0**1 = 2ra — 1. The differential d on M is defined as *
**d(A) = 0, d(vj) = h****t+j****, 1 < j < s, d(v****0****) = h****t**** - T****2n****_****k****, and d(u****0****) = o****k****T****n****_****k****. **
*Case 2a: *

*Let n = 2ra + 1, fc = 25, n — k = 2t + l, k < ra. Let M = A[v**x**,..., v**s**], where *

**\Vj\ = *(t + J) ~ 1, d(vj) = h****t+j****, 1 < j < 5, d(A) = 0. **

*Case 2b: *

*Let n = 2ra + 1, A; = 2s + 1, n — k = 2t, k < ra. Define M = A[v**0**,v**1**,... ,v**s**], *
**where |^-| = 4(t+j)-l, 0<j<s, d(A) = 0, d(v****0****) = h****t****-T****2n****_****k****, d(****Vj****) = h****t+j****, ****l<j <s. **

*Case 3: *

**Let n = 2ra + 2, A; = 2s + 1, n — k = 2t+ 1. Let M = A[v****0****,v****x****, ...,v****s****], where ****K-l = 4(t+j)-l, l<j<s, \v****0****\ = 2ra + l , d(.A) = 0, d(v****0****) = 0, d(^.) = h****t+J****, ****l<j<s. **

**MAIN THEOREM. Let 2 < k < [n/2]. With notation as above, the commu-***tative differential graded algebra M**nk** is the minimal model for the oriented *
*Grassmann manifold G**n k** . *

*The minimal model of G**nl** = S '*n _ 1*, the (n — 1)-sphere, is well known *
*(see [3]). Since G**nk** = G**nn**_**k**, the hypothesis that k < [n/2] is not a re-*
*striction. *

The paper is organized as follows: In §2 we recall a basic theorem needed in
*the construction of minimal models of homogeneous spaces. In §3 we prove the *
Main Theorem stated above and deduce the formality of the oriented Grassmann
*manifolds. In §4 we obtain results on nilpotence of flag manifolds (Theorem 6) *
*and the classifying space for the orthogonal group (Theorem 7). *

### 2. Minimal Models of homogeneous spaces

*Let G be a connected compact simple Lie group. Let H be a closed connected *
*subgroup of G. One has the following description of the minimal model of the *
*smooth homogeneous manifold G/H. Let T C G be a maximal torus in G *
*such that S := T C\ H is a maximal torus of H. Denote by W the Weyl group *
*of G with respect to T and by TV' the Weyl group of H with respect to S. *

*Let m = d i m T , and let r = dim S. The group W acts on T and hence on *
*the real cohomology algebra H* (BT\ R) of the classifying space of T which *
*is a polynomial algebra over R in m generators each having degree 2. The *
*IV-invariant subalgebra can be identified with the real cohomology algebra of *
*BG. The cohomology algebra H* (BG\ R) is a polynomial algebra R [ F*X*, . . . , F**m**] *
*in homogeneous elements F- having even degrees. (See B o r e l [2].) Similarly, *
*H* (BH-, R) = R [ x*x*, . . . , x**r**]. Let p: H* (BG] R) -» H* (BH; R) denote the map *
*induced by the inclusion H C G. Let f**i** = p(F**i**)**1** 1 < i < m. Now let C = *
*C(G, H) denote the differential graded algebra (d.g.a) H*(BH; R)[u**x**,..., u**m**], *
*where \u**{**\ = \f**{**\ — 1, and the differential d is defined as d(H*(BH]M)) = 0, *
*and d(i^) = f**{**, 1 < i < m. Note that since \u**t**\ is odd, graded commutativity *
*implies that u**{**u- = —u-u^ and, in particular, that u**2**- = 0. *

**T H E O R E M 1. **

*(i) (H. Cartan) With notation as above, the minimal model M**G/H** of G/H *
*is isomorphic to the minimal model of the d.g.a. ( C ( G , i f ) , d ) . *

*(ii) (Cf. [15; p. 317, Example (ii), (v)].) The space G/H is formal if for some *
*integer s, 1 < s < m, the sequence f**1**,...J**s** is a regular sequence in *
*H*(BH; R), and the elements /*5 + 1*, . . . , f**m** belong to the ideal generated *
*by f**l**, • • •, f**s**. *

A proof is sketched in [16; §4, Chapter 5].

**R e m a r k 2. It is known that the sequence f***1**... , /*m is a regular sequence in
*H* (BH\ R) when H is of maximal rank in G. Hence when H is of maximal *
*rank, G/H is formal. See [16; Chapter 5, Theorem 4.16]. *

**3. Minimal Model of G**

**3. Minimal Model of G**

**nk**In this section we prove the Main Theorem stated in the introduction. We
*apply Theorem 1 to the case G = SO(n), H = SO(k) x SO(n - k). Write *
*n = 2m or 2m + 1, and k = 2s or 2s + 1, n — k = 2t or 2t + 1, m,s,t being *
*integers. We shall assume that k > 2, since minimal models of spheres are well *
*known. We take T to be the standard maximal torus w*Thich consists of elements
' = [*i > • • • > U* e** SO(2m) C SO(n), t**j* G R, where

*t(e**{**) = cos(27r^.)e*i + s i n ^ T r ^ e - ^ ,

*(ci + 1) = " sin(27r^.)e. + cos(27rtj)e.+ 1,

*where i = 2j — 1, 1 < i < 2m. (Here the e**{* denote the standard basis of Rn .)
*When n is odd or k is even, H is of maximal rank, m . When n is even and *
*k is odd, S := H D T is of dimension r = s + t = m — 1. For w G W, and *
*t = [ £*l 3. . . , £m*] G T , w • £ G T is obtained by a permutation of the £. and *
*changing certain of the £ • to — t-. When n = 2m the number of sign changes *
*is to be even. From this it is easy to compute the W-invariant subalgebra of *
*H* (BT; R) = R [ ^ , . . . ,t**m**], |^. | = 2. Let P. denote the j th elementary symmet-*
*ric polynomial in t\,..., t**m**, and let v**m** = t**1**--t**m**. Then, if* (BSO(2m)\ R) = *
R [ PX, . . . , Pm_x, crm*], and H* ( £ S O ( 2 m + 1); R) = R [ P*X*, . . . , P**m**]. Note that *
*P**m** = a**m** G H*(B5'0(2m);R). The element ( - l )*rPr is the Pontrjagin class
*of the canonical n plane bundle over BSO(n)\ when n = 2m + 1 the (inte-*
gral) Euler class of the canonical bundle is of order 2 and hence it vanishes in
*real cohomology (cf. [12]). The calculation of H*(BH;R) is similar. One has *
*H*(BH]R) =R\p**l**,...,p**s**,q**l**,...,q**v**o**k**,T**n**_**k**], where a**k** = 0 (resp. a\ = p**s**) if *
*k is odd (resp. even), and r**n**_**k** = 0 (resp. T**n**_**k** = q**t**) when n — k is odd (resp. *

*even). The restriction map p: H*(BG;R) -* H*(BH,R) is given by *
*(i) p(P**r**) = £ PiQj^'fr* l<r<s + t, *

*i-\rj—r *

*(**... . . f < V T „ -*f c= : 0 if (n,fc) = (2m,2a)>

( H ) / , ( < T-) = j 0 otherwise.

*(It is understood that p**0** = q**0** = 1.) *

*It can be shown that when (n,k) = (2m, 2s) the elements 6, /*1 ?*. . . , f**m**_**l *

*form a regular sequence in the ring R := I7*(M;R). To see this, we note *

*that, R/(6) is isomorphic to the polynomial ring over R in p*

*lt*

### ... ,p

s*, q*

*x*

*,..., q*

*t*

*modulo the ideal generated by p*

*s*

*q*

*t*

* = f*

*m*

*. (This is because 0*

*2*

* = p*

*s*

*q*

*t*

*-) Since, * *by [3; Proposition 23.7], /-_,..., f*

*m*

### forms a regular sequence in the polynomial *algebra R\p*

*x*

### ,... ,p

5*, q*

*x*

*,..., q*

*t*

*] it follows that 0, f*

*v*

### . . . , /

m - 1### forms a regular *sequence in H*(BH;R). Theorem l(ii) shows in particular that the space G*

*nk*

*is formal when n and k are both even. Similarly it is seen that /-_,..., f*

*m*

### is *a regular sequence in H* (BH; R) when n = 2ra + 1. Tims we conclude that * *G*

*n k*

* is formal when n is odd or k is even. Note that H is of maximal rank * *in G except when n is even and k odd. Therefore the formality of G*

*nk*

### when *n is odd or k even follows from Remark 2. In any case, as remarked in the * *introduction, G*

^{n k}### is formal since it is a Riemannian symmetric space. We shall *verify formality of G*

*n k*

* directly for all values of n and k. *

*Let P = R\p*

*x*

*,... ,p*

*s*

*] be a polynomial algebra, where \pA = 4j, 1 < / < 5, * *and let h*

*r*

* G P be defined by (l + /i*

1### +/i

2### + -•- + /i

r* + -• •) = ( 1 + ^ + - • - + P J "*

1### , *where \hA = 4j. We have the following lemma: *

*LEMMA 3. For any non-negative integer t, the elements h*

*t+1J*

*... >h*

*t+s*

* form * *a regular sequence in the polynomial algebra P = R\p*

*1*

### ,... , p j .

*P r o o f . Let P = R\p*

*x*

*,..., p*

*s*

*]. When 8 = 1, the lemma is obviously true * *for any t. Assume inductively the statement holds for any t when s is replaced * *by s - 1. *

*By the induction hypothesis, for any t, h*

*t+1*

*,..., h*

*t+s*

*_*

*1*

### is a regular sequence *in P := P/(p*

*s*

*) = % ! , . . . , p*

5*_ J . Equivalent^, h*

*t+1*

*,..., h*

*t+s*

*_*

*1*

*,p*

*s*

### is a regular *sequence in P . In particular, p*

*s*

* mod (h*

*t+1*

*,..., h*

*t+s*

*_*

*1*

*) is not a zero divisor in * *P/(h*

*t+v*

*..., -Vs-i) •*

*N o t e t h a t h*

*t+s +*

*h*

*t+*

*s*

*-iPi + * *' +*

*h*

*tP*

*s*

### = 0 in P . Hence *h*

^{t+s}* = h*

^{t}*p*

^{s}* modulo the ideal (h*

^{t+1}*,..., ^*

^{t + s - 1}

### ) C P .

*When t = 0, it is clear that the ideal (h*

*t+1*

*,..., h*

*t+s*

### _

x*) = (p*

*1*

*,... ,p*

s### _-_) and /i

t + a### EE ^

0*p*

5* = p*

*s*

* is clearly not a zero divisor in P/(h*

*t+1*

*,..., h*

*t+s*

*_*

*1*

*) in this * *case. Assume that t > 1 and that the lemma holds when t is replaced by t — 1. *

### Hence ft

t*,..., Z^+s^ is a regular sequence. It follows that h*

*t*

### is not a zero divisor *in P/(h*

*t+1*

*,..., /**+,_!>. It follows that h*

*t+s*

* = h*

*t*

*p*

*s*

* modulo (h*

*i+1*

*,..., h*

*t+s*

*_*

*x*

*) *

*is not a zero divisor in P/(h*

*t+1*

*,..., / i*

t + 5 - 1*) . The lemma follows. • * We shall now establish the Main Theorem stated in the introduction.

### P r o o f of M a i n T h e o r e m . L e t 2 < f c < [n/2].

*Case 1: *

*Let n = 2ra, k = 2s, n — fc =- 2t, 1 < s < t. In this case the commuta-*

*tive d.g.a. C*

*ntk*

* := C(SO(n), SO(k) x SO(n- k)) is F ( M ; I ) [ w*

0### , . . . ,

V l### ] ,

*where d(H*(29H;R)) = 0 and du*

*r*

* = f*

*r*

* for 1 < r < ra, d(M*

0### ) = 5. Thus,

*writing u*

^{m}* = 9 • u*

^{0}*, one has du*

^{m}* = 6 • du*

^{0}* = 6*

^{2}* = p*

^{s}*q*

^{t}* =: /*

^{m}

### , and hence

*(l + du**1** + ... + du**m**) = (l + f**1** + ... + f**m**) = (l+p**1** + ...+p**3**)(l + q**l** + --- + q**t**). *

*Writing (1 + h**1** + ...) = (l+p**1**...p**3**)-\ one has h**r** = _T ff)(-l)'*a'pa,

l | a | l = r

*where a = ( a _ , . . . , a**3**) is a sequence of non-negative integers, ||a|| = ___) ia**i*,
Ial = _Ca»> ( a ) denotes the multinomial coefficient | a | ! / ( ax! • • •<-*_!), a n d Pa

**i **

*denotes the monomial \\ p?**{*. One has the following inhomogeneous equation
-<«<*

*in C**nk**: 1 + q**x** + • • • + q**t* = (1 + du_ + • • • + d um) ( l + /i_ + • • • ) . In particular
*we obtain, for 1 < j < s, *

*h**t+j = ~(**h**t+j-i** du**i + ""' +** h**i** du**t+j-i +** du**t+j) • *
*When n — k is even, g*t = r ^ ^ and hence

/it* - Tl_**k* = - ( / it_ _ du_ + • • • + d ut) .

*Let A = R [ P i , . . . , P . - i ^ f c ^ n - f c ] C fT*(B.ff;R). Let A 4*M = - 4 [ u0, t ;0, . . .

•••>va-_] denote the commutative d.g.a. over R , where |v.j = | ^t + J| — 1 =
*4(t + j) - 1, 0 < j < s, and |u*0| = |0| - 1 = 2ra - 1. T h e differential d
*on M**nk** is defined as follows: d(vj) = h*t + J*. , 1 < j < s , d(v**0**) = h**t* — r^_f c,
d(u0*) =a**k**T**n**_**k**, and d(.A) = 0. Clearly M**nk* is a free d.g.a. over R .

*Note that since t > s, / i*t*, • € A is decomposable for j > 1. Also, since *
*p**3** = cr**k**, h**3** e A is decomposable. It follows that M**n k* is minimal as a d.g.a.

*over R . We shall prove that M**n k** is a model for the d.g.a. C**nk**. From The-*
*orem 1, it will follow that M**nk** is a minimal model for G**nk**. *

*Let <\>: M**n k* -» C*n k** be the _4-algebra homomorphism defined by <f>(u**0**) = u*0,
*4>(vj) = -(h**t**_l**J**_**1**u**1**+... + u**tH**), l<j<s, <f>(v**0**) = - ( V*1u1 + --- + ut) . Then

*<£ is a morphism of d.g.a.'s. Indeed, d(<f>(u**0**)) = d(u*0*) = 0 = <f>(0) = <j)(d(u**0**)), *
*and, for 1 < j < _., one has d(<f>(vj)) = -d(h**t**_**rj**_**1**u**1* + .«• + ut + J.) =
*-(**h**t+f-i du**x** + ..-+ d u*t + i) = fcl+i* = 4>(h**i+j**) = </>(d(vj)), since d(h**r**) = 0 *
as / ir* e R [ p _ , . . . , p j C .A. Similarly, d ( # v*0*) ) = h**t* - r * , * = ^ ( d ( v0) ) . To show
*that the chain map <f> induces an isomorphism in cohomology, first observe that *
*H*(C**nk**,d) _. H*(G**nk**;R). This is because ( C*n i k, d ) is a model for the space
C*n k* (see Theorem 1). Alternatively, one applies a Koszul complex argument
(cf. [11; Chapter XXI, §4]) and uses the fact that # , / _ , . . • , /m_ _ is a regular
*sequence to see that the cohomology of C**n k** is H*(BH\ R ) / ( 0 , / _ , . . . , /*m_ _ ) =

*H*** iPn fc!*R) * U s i nS t h e relation ( 1 + / . + - • • + /m) ( l + ' »1+ - ' *) = ( 1 + 9 . + - ' •+<?_),
*we see that 0 = f**r* = _T Ptf; , 1 < r < <>i n* H*(BH;R)/(<T**k**T**n**_**k**, f^ .. . , /*m_ _ )

«+>=r

*- II* ( G „*i t; R) • In particular,* qj** = h**j** for any j , l<j<t and r*_*fc* = q**t** = h**t *

*in H*(BH;R)/(a**k**T**n**_**k**,f**v* , / J . Hence

**-**(G„,**

^{fe}**;R) =%!,••• , P , - i , ' t**

**;R) =%!,••• , P , - i , ' t**

^{i}**, v J / ( V i V . - i . V**

^{n}**- ^ - "T**

^{2}**) • **

*Using Lemma 3, one sees that T**n**_**k**, h**t** — T**n**_**k**,h**t+1**,..., h**t+8**_**1** and cr**k**,h**t** — *
*T**n**_**k**,h**t+1**,..., h**t+s**_**1** are regular sequences in A. From [3; Lemma 23.6] it fol-*
*lows that 6,T**n**_**k**—h**t**,h**t+1**,...,h**m**_**1** is a regular sequence in A. Again by ap-*
plying a Koszul complex argument we obtain that

*H*(M**n<k**,d) e. A/(**T**l_**k** - h**v**h**t+1**,...,h**m**_**x**,cr**k**T**n**_**k**) 9. H*(G**n**y,R) . *
*Under our identifications, the map (j) actually induces the identity map of *
*H*(G**n k**,R) . This proves that M** k** is quasi isomorphic to C**nk* and hence
*it is the minimal model of G** k* in this case.

*Case 2: *

*Let n = 2m + 1 = 2s + 2t + 1, k = 2s, or 25 + 1. We assume that fc < m *
*(equivalently s < t with equality only if fc = 2s). In this case C**nk* =

*H*(BH;R)[u*

*1*

*,...,u*

*m*

*], where d(H*(BH;R)) =0, duj-fj, 1 < j < m.' *

*Let A C H*(BH;R) be the polynomial algebra over R in generators *
P i r - . , PM, ^ (resp. P i , . . . , P5*, T*n*_ f e ) for fc even (resp.fc odd). Thus p**s* = ^
when fc = 25.

*Subcase (a): *

Let fc = 25. Let A tn fc* = A ^ , . . . ,v**s**] be the d.g.a. over R, where 1^1 = *

| / ij + t*| - l = 4 ( j + t ) - l and d ( ^ ) = h**t+j**, l<j<s, d(A) = 0. The free d.g.a. *

*M**nk** is minimal since the h**t+J**- are decomposable for j > 1. The A-algebra *

*map (j) : M**nk** -+ C**nk** defined by <f>(Vj) = -(hj^^u^ + • • • + u**j+t**), l<j<s, *
*is a morphism of d.g.a.'s. Also, the elements h**t+**j, 1 < j < s, form a regular *

*sequence in A. Therefore arguing as in case 1 above, we conclude that (j) is a *
*quasi isomorphism. Hence M**nk** is a minimal model of G**n k**. *

*Subcase (b): *

*Let fc = 25 + 1 . Let M**nk** = -4[t>*0,i'1*,... ,-uJ be the commutative d.g.a. over *
*R, where | ^ | = \h**t+j**\ - l = 4(t + j) - 1, 0 < j < s, and d(v**0**) = h**t** - T*_**k*,
*d(vA = h**t+j**, 1 < j < 5, and d(A) = 0. Since s < t, h**t+**- is decomposable *
*for j > 0. Therefore M**nk** is minimal. The A-algcbra map 6: M**nk** —> C**n k *

*defined by ^(Vj) = —(h**t+J**_**l**u**l** + • • • + Uj), 0 < j < s, is a morphism of d.g.a. *

*over R. Using the fact that h**t** — T^_**k**, h**i+1**,...,h**m* is a regular sequence in
*A = R\p**1**,... ,P**s**,T**n**_**k**], we conclude, as before, that M**n h* is a minimal model

*Case 3: *

*Let n = 2ra + 2, fc = 2$ + l , n — fc = 2£ + l , 1 < 5 < t . In this case the *
*subgroup SO(k) x SO(n — k) is not of maximal rank in SO(n). The c.d.g.a. *

*C**nk** has the description H*(*J*BH;E)[u*1,... , um, i i0*] with | i * | = A(t + j ) — 1, *

|ii0*1 = 2ra + 1 = n — 1, and du- = / •, 1 < j < ra, and du**0* = 0.

*Let A = P = % ! , . . . , p**s**] C H*(BH;R). Let yVf*nfc denote the d.g.a.

*A ^ , . . . , ^ , ^ ] , where \vj\ = A(t + j ) - 1, 1 < j < ' 5 , |u*0| = 2ra + 1,
d ( f ) = ^J + t, 1 < j < «s, df0* = 0, and d(A) = 0. As before, M**nk* is free
*and minimal. *

*The A-algebra map </>: M**n k** —r C**n k** defined by 4>(v-) = u - , 0 < j < 5, is a *
*chain map as can be verified as in Case 1. Note that the cohomology of M**n k* can
*again be computed using the Koszul complex of A with respect to the sequence *
/ it + 1*, . , . , / i*t + s*, 0 G A. Again using the fact that h**t+1**,... ,h**t**+**s* is a regular
sequence, a simple calculation leads to:

*H*(M**n**y,R) = A[u**0**}/(h**t+1**,...,h**t+s**). *

*This is also the cohomology of C**n k* and as in Case 1, we see that 0 induces
*isomorphism in cohomology. Hence M**nk** is the minimal model of G**n k**. *

*This completes the proof of the Main Theorem. D *
*COROLLARY 4 . The oriented Grassmannian G**n k** is formal for all 1 < k < n. *

*In particular all Massey products in H* (G**n k**] R) vanish. *

*P r o o f . First let n = 25 + 2t + 2, k = 25 + 1. With notation as above, *
*note that the A-algebra map M**nk** -> A[u**0**]/(h**t+1**,...,h**t**+**s**) = H*(M**n**y,R) *
*defined by v**0** i-> u**0**, v • i-> 0, 1 < j < s, is a map of d.g.a.'s where the differential *
*on H*(M**n k**]R) is defined to be zero. Hence G**nk* is formal.

*The same argument as above shows that G**n k** is formal for all parities of n *

*and k. • *
*COROLLARY 5. Let dim*Rv7rr(<5n^) cg)z* R) = 7 r*r.

*(i) Let n = 2s + 2t, k = 2s**}** l<s<t. *

*Then £ 7T**r**z**r** = 1 + z**n**~**l** + z**2s** + z**2t** + z**At**~**l** + £ (z^ + z*4^ ' ) "1) .

r > 0 l < j < 5

(ii)

*(a) Let n = 25 + 2t + 1, fc = 25, s<t. *

Tfen E v r = 1 + ^ + ^4 ( s + t H*+ £ (z*i + z**4**^ -**1**) . *

r > 0 l < j < 5

*(b) Let n = 2s + 2£ + 1. k = 25 + 1, s <t. *

*Then £ fr*rzr = 1 +* z**2t** + z**U**~**l* + £ (*4j + £4 ( i + j ) _ 1) .

r > 0 l < j < s

*(iii) Let n = 2s + 2t + 2. fc = 25 + 1, 1 < 5 < t. *

*Then £ n**r**z**r** = 1 + z**n**~**x* + £ (z4^' + z4^ ' ) -1) .

r > 0 l < j < 5

*_ P r o o f . This follows from the above description of the minimal model of *
*G**nk** and the fact that Hom**z**(7r**r**(G**n k**),R) is isomorphic to the r t h degree *
*component of the graded vector space M**nk**/V, where V denotes the ideal *

*M**nk**-M**nk** of "decomposable elements". • *

### 4. Nilpotence of Grassmannians and related spaces

*Let X be a path connected topological space with base point x. Recall that *
A" is called nilpotent if the fundamental group 7r = 7r1 (Ar*, x) of A*r is nilpotent
*as a group and all the higher homotopy groups of X are nilpotent as modules *
*over the integral group ring Zrr. That is, denoting the augmentation ideal of ZTT *
by 7, Ar* is nilpotent if and only if n is nilpotent and, for each n > 2, there *
*exists an integer N = N(n) such that I**N* • 7rn*(A, x) = 0. *

A path connected topological space Ar is said to be of finite Q-type if
*H**n**(X]Q) is finite dimensional for all n > 1. If X is nilpotent, then X is *
*of finite Q type if and only if H**X**(X; Q) and n**n**(X) ® Q are finite dimensional *
*for all n > 2. When X is a nilpotent space of finite Q-type, one associates to *
Ar* the minimal model M**x** of the Sullivan-de Rham complex of X which is a *
*c.d.g.a. over Q. The minimal model M**x** contains all the rational homotopy *
*information of X in this case. That is, M**x** and M**Y* are quasi isomorphic if
and only if Ar* and Y are of same rational homotopy type, where both X and Y *
are of finite Q-type. We refer the reader to [1; Chapter 2] for details. (See also
[4; §9].) In case Ar is a smooth manifold, it is more convenient to work with the
*real homotopy theory via the minimal model of the de Rham complex of X. *

*Let n j , . . . , n**s** be a sequence of positive integers, and let n = ^2** n**i • Denote *

*l<i<s *

by Ar* = G(n**x**,..., n**s**) the flag manifold consisting of flags ( V*1 5*. . . , V**s**), where *

*\] is an n**{** dimensional vector subspace of R**n** such that V**{** ± V- if i ^ j , *
*and \\ 0 • • • © V**s** = R**n** . The flag manifold X can be identified with the coset *
*space 0(n)/(0(n**x**) x ••• x 0(n**s**)) so that it is naturally a smooth compact *
*manifold of dimension ^ n**{**n •. When s = 2, it is identified with the Grass-*

*l<i<j<s *

*mannian G**nni*. The universal covering of the flag manifold is the oriented flag
manifold Ar* = G(n**x**,... , n j = SO(n)/'(50(71,) x • • • x SO(n**s**)), which consists *
of "oriented flags", that is, flags ( V1 ?*. . . , V**s**) together with orientations on each *
*vector space V**{** so that the direct sum orientation on ^ V**{* = lRn coincides with
the standard orientation on En* ._The natural map p: X -> X that forgets the *
orientations on oriented flags of Ar* is the covering projection. The deck transfor-*
*mation group is generated by the involutions a •, 1 < i < s, which reverse the *
orientation on zth and 5 th vector space in each oriented flag, and is isomorphic

to ( Z / 2 )5 - 1*. We note that, when n**{** and n**s** are odd, a**{* can be realized as
*multiplication on the left by the element A**{** € SO(n), where A**{* has, as a block
*diagonal matrix j th block down the diagonal the identity matrix of size n • if *
*j ^ i,s and the zth and 5th block being negative identity matrices of sizes n**{ *

*and n**s* respectively.

In this section we prove the following theorem:

*THEOREM 6. ( G l o v e r — H o m e r [6]) Let X = G(n**x**,... ,n**3**), s > 2, de-*
*note the real flag manifold. Then the following are equivalent: *

*(i) all the n**{** are odd, *
*(ii) X is simple, *
*(iii) X is nilpotent. *

*THEOREM 7. (Varadarajan) The classifying space BO(k) is nilpotent if and *
*only if k is odd. *

We need the following lemma (cf. [14; Chapter 7, §3, Lemma 7]). We shall
*denote the free homotopy classes from the n-sphere to Y by 7r*n(F). Note that
*when Y is a simply connected space, n**n**{Y) may be identified with 7r*n(F, y ) .
*LEMMA 8. Let p: X —•> X be the universal covering projection of a path con-*
*nected finite CW complex, and let r > 2 . Then p induces an isomorphism *
*between TT (X) and 7r**r**(X,x), which is compatible with the action of the deck *
*transformation group on n**r**(X) and that of** TT**X**(X,X)** on ir**r**(X,x). *

P r o o f of T h e o r e m 6.

*(i) -==> (ii): Assume that all the n**{** are odd. Since SO(n) is connected, *
*maps induced on X by multiplication by elements of SO(n) are all homotopic *
*to the identity map. In particular the a**t**, 1 < i < s, are homotopic to the *
*identity map of X. Since the a • generate the deck transformation group of the *
*universal covering projection X —> X, it follows from Lemma 8 that the action *
*of n**1**(X) = ( Z / 2 )*s _ 1 on any 7rr*(X) is trivial and so the space X is simple-*

It is evident that (ii) implies (iii).

*(iii) =-> (i): We will first assume that 5 = 2 so that X is the Grassmann *
*manifold X = G**nn** . For simplicity of notation let k = n**x**. Assume that at *
*least one of the integers k,n—k is even. We will prove that X is not nilpotent. *

*Indeed, let r be an even integer in the set {k,n — k}. We will prove that the *
action of the generator of* TT**1**(X)** = Z / 2 on 7T*r(X)®zR has —1 as an eigenvalue.

*When r = n — k > k it will be convenient to denote by a**r* the element that
*was denoted T**n**_**k* in §3.

*We first note that the deck transformation a := a**x* of the covering projection
*G**n k** —r G**n k** is not homotopic to the identity. In fact a reverses the orientation *

*on the canonical k-plane bundle over X = G , . When r is even, the Euler *
*class a**r** of the canonical oriented r -plane bundle is not torsion and in real *
*cohomology one has a*(a**r**) = —o**r**. The map a induces an endomorphism a *
*of the d.g.a. M**n k** which is unique up to chain homotopy. The map a induces *
an involution [a], which depends only on the chain homotopy class of d , of
*the graded R-vector space M**nk**jV such that the following diagram commutes, *
*where the vertical arrows are natural isomorphisms (cf. [5]). Here a* denotes *
the R-linear involution on Homz(7rr(Gn fc*),R) induced by a. *

*(M„,JVY -!=L> (M„*

^{:}*JVY *

**I I **

Homz(7rr(Gnife),IR) - ^ - > Homz(7rr(Gn > f e),E)

Note that by Corollary 5 the vector space Homz(7rr(Gn fc),R) is non-zero. In
fact the class crr* is not in the ideal V of decomposable elements of M**n k**. Also *
[a](crr*) = — cr*r* since in de Rham cohomology a* maps the Euler class of the *
*canonical r-plane bundle to its negative. Hence it follows that the —1 eigenspace *
*of a*: Hom*z(7rr(Gn A;*),R) —> Hom**z**(7r**r**(G**n fc*),R) is non-zero. It follows that

— 1 is an eigenvalue for the action of the generator of the fundamental group of

*G**n,k* o n* *A**G**n,k) ® z*R-* H e n c e G**n,k* i s n o t nilpotent.

*Now let s > 3 and assume without loss of generality that n*x is even.

*One has a natural inclusion j : Y := G(n**x**,n**2**) —r X and a projection map *
*q: X -> G{n^n**2** + ••• + n**8**) =: Z. Explicitly, j(A) = (A,E**2**,...,E**8**) and *
*q(\\,..., V**s**) = V**l**, where E**r* denotes the span of the standard basis vectors et,
*n**x**-\ h n**r**_**x** + 1 <t <n**x**-\ \-n**r**. Note that qoj is the natural inclusion *
*of Y into Z induced by the inclusion of R*n- +n- into Rn and hence induces
*an isomorphism of fundamental groups. Denote by f:Y—>Z a lift of q o j o p *
*to Z := G*n n i + n 2*, where p: Y := G*n i + n 2 ? n i* -> Y is the universal covering *
*projection. The map / pulls back the canonical n**x** -plane bundle on Z to that *
*on Y. Hence it maps the Euler class a**n** (Z) to the class +a**n** (Y). Replacing *
*/ by / o a if necessary one may as well assume that / * (a**n** (Z)) = a**ni** (Y). As *
in the case when 5 = 2, using naturality properties of minimal models, one con-
*cludes that the morphism of d.g.a. induced by / maps the class a (Z) G Mg to *
*O~*n* (Y) 6 My • This implies that the —1 eigenspace for the action of n^Y) (via *
*the isomorphism of fundamental groups induced by qoj) on Hom(7r*n (Z),R)
is non-zero. Hence the action of 7r1* (Y) via the monomorphism of fundamental *
*groups induced by j on Hom(7r*n (X),R) must have non-zero - 1 eigenspace.

*This clearly implies that the action of ^**X**(X) on 7r (X) is not nilpotent. • *

R e m a r k 9. Theorem 7.2 of [10] can be readily applied to show that (iii) = > (i)
once this is known for Grassmannians. Our proof, however, exhibits a higher
*homotopy group of X on which the action of the fundamental group is not *
nilpotent. Alternatively, one can deduce the same result from our result for
Grassmannians using the second Claim in the proof of [10; Theorem 7.2]. We
record, as a corollary of the the above proof, the following proposition for pos-
sible future reference.

*PROPOSITION 10. Let X = G(n**v**...,n**s**), s > 2 . If n**{** is an even integer, *
*then it**n**.(X) is not nilpotent as a 7r*1*(X) module. *

*P r o o f of T h e o r e m 7. Note that BO(k) — |J G**n k*, where we regard

*n>2k *

*G**n k* as the subspace of G*n + 1 k* in the usual way, considering the vector space
*R**n* as the subspace of Mn + 1 consisting of those vectors with last coordinate
*being zero. The inclusion map i**n**: G**n k** —» BO(k) is an (n — k) -equivalence and *
*hence, given any r > l , f o r n > k + r + l , the map i**n* induces isomorphism of
*the r th homotopy groups. Also the action of the fundamental group of G**n k* on
*n**r**(G**n k**,x) is compatible with the action of the fundamental group of BO(k) *
*on 7r**r**(BO(k), x) via the map induced by i**n**. In particular if k is even, choosing *
*r — k and n > 2k + 1, one sees from our proof of Theorem 6 that BO(k) is not *
nilpotent.

*When k is odd, one can always choose n to be even (in addition to n > *

*k + r + 1) to conclude that the fundamental group of BO(k) acts trivially on *
*n**r**(G**n k**) for any r > 1. Hence we conclude that the space BO(k) is nilpotent. *

*In fact we have shown that BO(k) is simple. D *

**Acknowledgement **

We thank K. Varadarajan for his interest in this problem and for making available to us his paper [17]. We thank J. Korbas for his comments on an earlier version of this paper and, in particular, for pointing out to us the work of H. G l o v e r and W. H o m e r [6].

R E F E R E N C E S

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[12] MILNOR, J . — S T A S H E F F , J . : Characteristic Classes. Ann. of Math. Stud. 76, Princeton Univ Press, Princeton, N J , 1974.

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[14] SPANIER, E . : Algebraic Topology, Springer-Verlag, New York, 1979.

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[16] TRALLE, A . — O P R E A , J . : Symplectic Manifolds with no Kähler Structure. Lecture Notes in M a t h . 1661, Springer Verlag, New York.

*[17] V A R A D A R A J A N , K . : Nilpotent actions and nilpotent spaces, 1976 (Unpublished). *

*Received December 1, 1998 * Stat-Math Unit *

*Revised Anril 15 1999 Indian Statistical Institute *
*203 Barrackpore Trunk Road *
*Calcutta 700 035 *

*INDIA *

*** Chennai Mathematical Institute *
*92 G.N. Chetty Road *

*Chennai 600 017 *
*INDIA *

*E-mail: goutam@isical.ac.in *
sankaran@smi.eгnet.in